Riemannian manifold

noun2

1. •
   A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;

   • In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded. — 1984
2. •
   an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.